This allows the site administrator to specify an email address that the wiki can contact if something goes wrong.
Current versions of MediaWiki (1.13 and above) do not use this setting, having delegated its uses to $wgPasswordSender, but future versions or extensions might.

The installer automatically generates a value for this setting, and places it towards the top of LocalSettings.php.
The generated value will often be nonsensical ("apache@localhost" or similar), so be sure to change $wgEmergencyContact and $wgPasswordSender if they are incorrect like this.